reserve x,N for set,
        k for Nat;
reserve N for with_zero set;
reserve S for IC-Ins-separated non empty with_non-empty_values
     Mem-Struct over N;

theorem Th3:
  for N for S being Mem-Struct over N
  for p being PartState of S holds not IC S in dom DataPart p
proof
  let N; let S be Mem-Struct over N;
  let p be PartState of S;
  assume
A1: IC S in dom DataPart p;
  dom DataPart p c= (the carrier of S) \ {IC S} by RELAT_1:58;
  then not IC S in {IC S} by A1,XBOOLE_0:def 5;
  hence contradiction by TARSKI:def 1;
end;
